We recall from [1]} that a justification of the BB steps is the fact that they approximate the Hessian matrix by a scalar multiple of the identity, as follows. It is reasonable that an approximation \(B_k\)  to the Hessian approximately satisfies the secant equation \(\mathbf {y}_{k-1} = B_k \mathbf {s}_{k-1}\) .
The BB steps solve the secant equation in a least-squares sense [1]}:
\( \alpha _{k}^{\rm {BB1}} = \operatornamewithlimits{\arg \!\min }_\alpha \Vert \mathbf {y}_{k-1} - \alpha \, \mathbf {s}_{k-1}\Vert , \qquad \alpha _{k}^{\rm {BB2}} = \operatornamewithlimits{\arg \!\min }_\alpha \Vert \mathbf {s}_{k-1} - \alpha ^{-1} \, \mathbf {y}_{k-1}\Vert ,\) 
